To indicate whether the Axiom of choice must be employed to select one element from each set in the following collections.

Question

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Answer 1

Question

Chapter 5.5, Problem 1E

a.

To determine

To indicate whether the Axiom of choice must be employed to select one element from each set in the following collections.

a.

Expert Solution

Explanation of Solution

Given:

Consider the infinite collection of sets, each set containing one odd and one even integer.

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

b.

To determine

To indicate whether the Axiom of Choice must be employed.

b.

Expert Solution

Explanation of Solution

Given:

Consider the finite collection of sets such that each set is uncountable

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

c.

To determine

To indicate whether the Axiom of Choice must be employed.

c.

Expert Solution

Explanation of Solution

Given:

Consider the infinite collection of sets each containing exactly four natural numbers.

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

d.

To determine

To indicate whether the Axiom of Choice must be employed.

d.

Expert Solution

Explanation of Solution

Given:

Consider the denumerable collection of uncountable sets.

Calculation:

Since, the collection is denumerable collection of uncountable sets.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

e.

To determine

To indicate whether the Axiom of Choice must be employed.

e.

Expert Solution

Explanation of Solution

Given:

Consider the collection of following sets-

{A:πℕ⊆A⊆ℝ−ℚ}

Where πℕ={πn:n∈ℕ}

Calculation:

Since the set of natural numbers are infinite.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

f.

To determine

To indicate whether the Axiom of Choice must be employed.

f.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℕ and both A and ℕ−A are infinite}

Calculation:

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

g.

To determine

To indicate whether the Axiom of Choice must be employed.

g.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℝ and both A and ℝ−A are infinite}

Calculation:

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

h.

To determine

To indicate whether the Axiom of Choice must be employed.

h.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℝ and A is denumerable}

Calculation:

Since, in the collection of sets, each set in A is infinite.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

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Answer 2

Question

Chapter 5.5, Problem 1E

a.

To determine

To indicate whether the Axiom of choice must be employed to select one element from each set in the following collections.

a.

Expert Solution

Explanation of Solution

Given:

Consider the infinite collection of sets, each set containing one odd and one even integer.

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

b.

To determine

To indicate whether the Axiom of Choice must be employed.

b.

Expert Solution

Explanation of Solution

Given:

Consider the finite collection of sets such that each set is uncountable

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

c.

To determine

To indicate whether the Axiom of Choice must be employed.

c.

Expert Solution

Explanation of Solution

Given:

Consider the infinite collection of sets each containing exactly four natural numbers.

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

d.

To determine

To indicate whether the Axiom of Choice must be employed.

d.

Expert Solution

Explanation of Solution

Given:

Consider the denumerable collection of uncountable sets.

Calculation:

Since, the collection is denumerable collection of uncountable sets.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

e.

To determine

To indicate whether the Axiom of Choice must be employed.

e.

Expert Solution

Explanation of Solution

Given:

Consider the collection of following sets-

{A:πℕ⊆A⊆ℝ−ℚ}

Where πℕ={πn:n∈ℕ}

Calculation:

Since the set of natural numbers are infinite.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

f.

To determine

To indicate whether the Axiom of Choice must be employed.

f.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℕ and both A and ℕ−A are infinite}

Calculation:

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

g.

To determine

To indicate whether the Axiom of Choice must be employed.

g.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℝ and both A and ℝ−A are infinite}

Calculation:

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

h.

To determine

To indicate whether the Axiom of Choice must be employed.

h.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℝ and A is denumerable}

Calculation:

Since, in the collection of sets, each set in A is infinite.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

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Answer 3

Question

Chapter 5.5, Problem 1E

a.

To determine

To indicate whether the Axiom of choice must be employed to select one element from each set in the following collections.

a.

Expert Solution

Explanation of Solution

Given:

Consider the infinite collection of sets, each set containing one odd and one even integer.

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

b.

To determine

To indicate whether the Axiom of Choice must be employed.

b.

Expert Solution

Explanation of Solution

Given:

Consider the finite collection of sets such that each set is uncountable

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

c.

To determine

To indicate whether the Axiom of Choice must be employed.

c.

Expert Solution

Explanation of Solution

Given:

Consider the infinite collection of sets each containing exactly four natural numbers.

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

d.

To determine

To indicate whether the Axiom of Choice must be employed.

d.

Expert Solution

Explanation of Solution

Given:

Consider the denumerable collection of uncountable sets.

Calculation:

Since, the collection is denumerable collection of uncountable sets.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

e.

To determine

To indicate whether the Axiom of Choice must be employed.

e.

Expert Solution

Explanation of Solution

Given:

Consider the collection of following sets-

{A:πℕ⊆A⊆ℝ−ℚ}

Where πℕ={πn:n∈ℕ}

Calculation:

Since the set of natural numbers are infinite.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

f.

To determine

To indicate whether the Axiom of Choice must be employed.

f.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℕ and both A and ℕ−A are infinite}

Calculation:

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

g.

To determine

To indicate whether the Axiom of Choice must be employed.

g.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℝ and both A and ℝ−A are infinite}

Calculation:

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

h.

To determine

To indicate whether the Axiom of Choice must be employed.

h.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℝ and A is denumerable}

Calculation:

Since, in the collection of sets, each set in A is infinite.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

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Answer 4

Question

Chapter 5.5, Problem 1E

a.

To determine

To indicate whether the Axiom of choice must be employed to select one element from each set in the following collections.

a.

Expert Solution

Explanation of Solution

Given:

Consider the infinite collection of sets, each set containing one odd and one even integer.

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

b.

To determine

To indicate whether the Axiom of Choice must be employed.

b.

Expert Solution

Explanation of Solution

Given:

Consider the finite collection of sets such that each set is uncountable

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

c.

To determine

To indicate whether the Axiom of Choice must be employed.

c.

Expert Solution

Explanation of Solution

Given:

Consider the infinite collection of sets each containing exactly four natural numbers.

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

d.

To determine

To indicate whether the Axiom of Choice must be employed.

d.

Expert Solution

Explanation of Solution

Given:

Consider the denumerable collection of uncountable sets.

Calculation:

Since, the collection is denumerable collection of uncountable sets.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

e.

To determine

To indicate whether the Axiom of Choice must be employed.

e.

Expert Solution

Explanation of Solution

Given:

Consider the collection of following sets-

{A:πℕ⊆A⊆ℝ−ℚ}

Where πℕ={πn:n∈ℕ}

Calculation:

Since the set of natural numbers are infinite.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

f.

To determine

To indicate whether the Axiom of Choice must be employed.

f.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℕ and both A and ℕ−A are infinite}

Calculation:

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

g.

To determine

To indicate whether the Axiom of Choice must be employed.

g.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℝ and both A and ℝ−A are infinite}

Calculation:

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

h.

To determine

To indicate whether the Axiom of Choice must be employed.

h.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℝ and A is denumerable}

Calculation:

Since, in the collection of sets, each set in A is infinite.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

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Answer 5

Chapter 5.5, Problem 1E

a.

To determine

To indicate whether the Axiom of choice must be employed to select one element from each set in the following collections.

a.

Expert Solution

Explanation of Solution

Given:

Consider the infinite collection of sets, each set containing one odd and one even integer.

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

b.

To determine

To indicate whether the Axiom of Choice must be employed.

b.

Expert Solution

Explanation of Solution

Given:

Consider the finite collection of sets such that each set is uncountable

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

c.

To determine

To indicate whether the Axiom of Choice must be employed.

c.

Expert Solution

Explanation of Solution

Given:

Consider the infinite collection of sets each containing exactly four natural numbers.

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

d.

To determine

To indicate whether the Axiom of Choice must be employed.

d.

Expert Solution

Explanation of Solution

Given:

Consider the denumerable collection of uncountable sets.

Calculation:

Since, the collection is denumerable collection of uncountable sets.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

e.

To determine

To indicate whether the Axiom of Choice must be employed.

e.

Expert Solution

Explanation of Solution

Given:

Consider the collection of following sets-

{A:πℕ⊆A⊆ℝ−ℚ}

Where πℕ={πn:n∈ℕ}

Calculation:

Since the set of natural numbers are infinite.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

f.

To determine

To indicate whether the Axiom of Choice must be employed.

f.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℕ and both A and ℕ−A are infinite}

Calculation:

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

g.

To determine

To indicate whether the Axiom of Choice must be employed.

g.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℝ and both A and ℝ−A are infinite}

Calculation:

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

h.

To determine

To indicate whether the Axiom of Choice must be employed.

h.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℝ and A is denumerable}

Calculation:

Since, in the collection of sets, each set in A is infinite.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

Answer 6

Chapter 5.5, Problem 1E

a.

To determine

To indicate whether the Axiom of choice must be employed to select one element from each set in the following collections.

a.

Expert Solution

Explanation of Solution

Given:

Consider the infinite collection of sets, each set containing one odd and one even integer.

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

Answer 7

Chapter 5.5, Problem 1E

a.

To determine

To indicate whether the Axiom of choice must be employed to select one element from each set in the following collections.

Answer 8

a.

Expert Solution

Explanation of Solution

Given:

Consider the infinite collection of sets, each set containing one odd and one even integer.

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

Answer 9

a.

Expert Solution

Answer 10

a.

Expert Solution

Answer 11

Expert Solution

Answer 12

b.

To determine

To indicate whether the Axiom of Choice must be employed.

b.

Expert Solution

Explanation of Solution

Given:

Consider the finite collection of sets such that each set is uncountable

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

Answer 13

b.

To determine

To indicate whether the Axiom of Choice must be employed.

Answer 14

b.

Expert Solution

Explanation of Solution

Given:

Consider the finite collection of sets such that each set is uncountable

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

Answer 15

b.

Expert Solution

Answer 16

b.

Expert Solution

Answer 17

Expert Solution

Answer 18

c.

To determine

To indicate whether the Axiom of Choice must be employed.

c.

Expert Solution

Explanation of Solution

Given:

Consider the infinite collection of sets each containing exactly four natural numbers.

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

Answer 19

c.

To determine

To indicate whether the Axiom of Choice must be employed.

Answer 20

c.

Expert Solution

Explanation of Solution

Given:

Consider the infinite collection of sets each containing exactly four natural numbers.

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

Answer 21

c.

Expert Solution

Answer 22

c.

Expert Solution

Answer 23

Expert Solution

Answer 24

d.

To determine

To indicate whether the Axiom of Choice must be employed.

d.

Expert Solution

Explanation of Solution

Given:

Consider the denumerable collection of uncountable sets.

Calculation:

Since, the collection is denumerable collection of uncountable sets.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

Answer 25

d.

To determine

To indicate whether the Axiom of Choice must be employed.

Answer 26

d.

Expert Solution

Explanation of Solution

Given:

Consider the denumerable collection of uncountable sets.

Calculation:

Since, the collection is denumerable collection of uncountable sets.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

Answer 27

d.

Expert Solution

Answer 28

d.

Expert Solution

Answer 29

Expert Solution

Answer 30

e.

To determine

To indicate whether the Axiom of Choice must be employed.

e.

Expert Solution

Explanation of Solution

Given:

Consider the collection of following sets-

{A:πℕ⊆A⊆ℝ−ℚ}

Where πℕ={πn:n∈ℕ}

Calculation:

Since the set of natural numbers are infinite.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

Answer 31

e.

To determine

To indicate whether the Axiom of Choice must be employed.

Answer 32

e.

Expert Solution

Explanation of Solution

Given:

Consider the collection of following sets-

{A:πℕ⊆A⊆ℝ−ℚ}

Where πℕ={πn:n∈ℕ}

Calculation:

Since the set of natural numbers are infinite.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

Answer 33

e.

Expert Solution

Answer 34

e.

Expert Solution

Answer 35

Expert Solution

Answer 36

f.

To determine

To indicate whether the Axiom of Choice must be employed.

f.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℕ and both A and ℕ−A are infinite}

Calculation:

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

Answer 37

f.

To determine

To indicate whether the Axiom of Choice must be employed.

Answer 38

f.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℕ and both A and ℕ−A are infinite}

Calculation:

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

Answer 39

f.

Expert Solution

Answer 40

f.

Expert Solution

Answer 41

Expert Solution

Answer 42

g.

To determine

To indicate whether the Axiom of Choice must be employed.

g.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℝ and both A and ℝ−A are infinite}

Calculation:

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

Answer 43

g.

To determine

To indicate whether the Axiom of Choice must be employed.

Answer 44

g.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℝ and both A and ℝ−A are infinite}

Calculation:

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

Answer 45

g.

Expert Solution

Answer 46

g.

Expert Solution

Answer 47

Expert Solution

Answer 48

h.

To determine

To indicate whether the Axiom of Choice must be employed.

h.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℝ and A is denumerable}

Calculation:

Since, in the collection of sets, each set in A is infinite.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

Answer 49

h.

To determine

To indicate whether the Axiom of Choice must be employed.

Answer 50

h.

Expert Solution

Explanation of Solution

Given:

Consider the following collection of sets

{A:A⊆ℝ and A is denumerable}

Calculation:

Since, in the collection of sets, each set in A is infinite.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if Α is any collection of nonempty sets, then there exists a function F from Α to ∪A∈ΑA such that for every A∈Α , F(A)∈A